Einstein's Diffusion and Probability-Wave Equations of Random ...
Einstein's Diffusion and Probability-Wave Equations of Random ...
Einstein's Diffusion and Probability-Wave Equations of Random ...
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Gauge Institute Journal,<br />
H. Vic Dannon<br />
3.<br />
Integral <strong>of</strong> Hyper-real Function<br />
In [Dan3], we defined the integral <strong>of</strong> a Hyper-real Function.<br />
Let f () x be a hyper-real function on the interval [ ab] , .<br />
The interval may not be bounded.<br />
f () x may take infinite hyper-real values, <strong>and</strong> need not be<br />
bounded.<br />
At each<br />
a<br />
≤<br />
x<br />
≤b,<br />
there is a rectangle with base<br />
dx<br />
[ x − , x + 2<br />
], height f () x ,<br />
dx<br />
2<br />
<strong>and</strong> area<br />
f ( xdx. )<br />
We form the Integration Sum <strong>of</strong> all the areas for the x ’s<br />
that start at x = a, <strong>and</strong> end at x = b,<br />
∑ f ( xdx ) .<br />
x∈[ a, b]<br />
If for any infinitesimal dx , the Integration Sum has the<br />
same hyper-real value, then f () x is integrable over the<br />
interval [ ab] , .<br />
12